Date of download: 1/23/2018 Copyright © ASME. All rights reserved. From: Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method J. Vib. Acoust. 2017;139(5):051010-051010-13. doi:10.1115/1.4036469 Figure Legend: (a) Geometry of a multilayered periodic composite, whose unit cell is spanned by primitive lattice vectors a1 and a3 in a two-dimensional (2D) coordinate space. Note that a1=||a1|| and a3=||a3||. (b) The corresponding wavevector space, where the topmost rectangle delineated by a thick solid line represents first Brillouin zone. Note that the aliasing paths are denoted by thick dotted lines.

Date of download: 1/23/2018 Copyright © ASME. All rights reserved. From: Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method J. Vib. Acoust. 2017;139(5):051010-051010-13. doi:10.1115/1.4036469 Figure Legend: (a) (Left) A unit cell of the infinitely periodic three-layered composite having a1/a3 = 2.0, which is employed for numerical dispersion analysis. (Right) The corresponding wavevector domain, which illustrates the valid wavevector path for waves perpendicular to the layers (Γ–X) and the corresponding aliasing paths (Γ′−X′, Γ″−X″). (b) FE dispersion relation obtained by employing a unit cell of a1/a3 = 2.0. Note that the numerical dispersion relation contains unwanted fictitious modes represented by red solid lines. (c) Three analytical dispersion relations obtained from Eq. (19): (c-1) κ1 = 0, (c-2) κ1 = 2π/a1, and (c-3) κ1 = 4π/a1. (d) The projection of all the analytical dispersion relations in Fig. 2(c) onto the κ3 − ω plane. Note that one can find one-to-one map in all the observed wave modes in Figs. 2(b) and 2(d), indicating that the fictitious modes originate from the aliasing paths. Moreover, the maximum valid frequency ωmax from Eq. (44) is denoted by a line with square markers.

Date of download: 1/23/2018 Copyright © ASME. All rights reserved. From: Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method J. Vib. Acoust. 2017;139(5):051010-051010-13. doi:10.1115/1.4036469 Figure Legend: (a) (Left) A unit cell of the infinitely periodic three-layered composite having a1/a3 = 1.0, which is employed for numerical dispersion analysis. (Right) The corresponding wavevector domain, which illustrates the valid wavevector path for waves perpendicular to the layers (Γ–X) and the corresponding aliasing paths (Γ′−X′, Γ″−X″). (b) FE dispersion relation obtained by employing a unit cell of a1/a3 = 1.0. Note that the numerical dispersion relation contains unwanted fictitious modes represented by red solid lines. (c) Three analytical dispersion relations obtained from Eq. (19): (c-1) κ1 = 0, (c-2) κ1 = 2π/a1, and (c-3) κ1 = 4π/a1. (d) The projection of all the analytical dispersion relations in Fig. 3(c) onto the κ3 − ω plane. Note that one can find one-to-one map in all the observed wave modes in Figs. 3(b) and 3(d), indicating that the fictitious modes originate from the aliasing paths. Moreover, the maximum valid frequency ωmax from Eq. (44) is denoted by a line with square markers.

Date of download: 1/23/2018 Copyright © ASME. All rights reserved. From: Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method J. Vib. Acoust. 2017;139(5):051010-051010-13. doi:10.1115/1.4036469 Figure Legend: (a) (Left) A unit cell of the infinitely periodic four-layered composite having a1/a3 = 2.0, which is employed for numerical dispersion analysis. (Right) The corresponding wavevector domain, which illustrates the valid wavevector path for waves perpendicular to the layers (Γ – X) and the corresponding aliasing paths (Γ′−X′, Γ″−X″). (b) FE dispersion relation obtained by employing a unit cell of a1/a3 = 2.0. Note that the numerical dispersion relation contains unwanted fictitious modes represented by red solid lines. (c) Three analytical dispersion relations obtained from Eq. (19): (c-1) κ1 = 0, (c-2) κ1 = 2π/a1, and (c-3) κ1 = 4π/a1. (d) The projection of all the analytical dispersion relations in Fig. 4(c) onto the κ3 − ω plane. Note that one can find one-to-one map in all the observed wave modes in Figs. 4(b) and 4(d), indicating that the fictitious modes originate from the aliasing paths. Moreover, the maximum valid frequency ωmax from Eq. (44) is denoted by a line with square markers.

Date of download: 1/23/2018 Copyright © ASME. All rights reserved. From: Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method J. Vib. Acoust. 2017;139(5):051010-051010-13. doi:10.1115/1.4036469 Figure Legend: (a) (Left) A unit cell of the infinitely periodic four-layered composite having a1/a3 = 1.0, which is employed for numerical dispersion analysis. (Right) The corresponding wavevector domain, which illustrates the valid wavevector path for waves perpendicular to the layers (Γ–X) and the corresponding aliasing paths (Γ′−X′, Γ″−X″). (b) FE dispersion relation obtained by employing a unit cell of a1/a3 = 1.0. Note that the numerical dispersion relation contains unwanted fictitious modes represented by red solid lines. (c) Three analytical dispersion relation obtained from Eq. (19): (c-1) κ1 = 0, (c-2) κ1 = 2π/a1, and (c-3) κ1 = 4π/a1. (d) The projection of all the analytical dispersion relations in Fig. 5(c) onto the κ3 − ω plane. Note that one can find one-to-one map in all the observed modes in Figs. 5(b) and 5(d), indicating that the fictitious modes originate from the aliasing paths. Moreover, the maximum valid frequency ωmax from Eq. (44) is denoted by a line with square markers.

Date of download: 1/23/2018 Copyright © ASME. All rights reserved. From: Generalized Spatial Aliasing Solution for the Dispersion Analysis of Infinitely Periodic Multilayered Composites Using the Finite Element Method J. Vib. Acoust. 2017;139(5):051010-051010-13. doi:10.1115/1.4036469 Figure Legend: (a) Analytical dispersion relation obtained from Eq. (19) for the infinitely periodic three-layered composite: (a-1) κ3 = 0 and (a-2) κ3 = π/a3. (b) Analytical dispersion relation obtained from Eq. (19) for the infinitely periodic four-layered composite: (b-1) κ3 = 0 and (b-2) κ3 = π/a3. The dotted lines denote the approximated linear dispersion relations obtained from the effective modulus theory (41). Note that the κ1 axis is intentionally normalized by a3 because the periodic length a3 is the common characteristic length of the considered periodic composites.